Friedel oscillations of one-dimensional correlated fermions from perturbation theory and density functional theory
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THE EUROPEAN PHYSICAL JOURNAL B
Regular Article
Friedel oscillations of one-dimensional correlated fermions from perturbation theory and density functional theory Jovan Odavi´c1,a , Nicole Helbig 1,2,3 , and Volker Meden 1 1
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Institut f¨ ur Theorie der Statistischen Physik, RWTH Aachen University and JARA – Fundamentals of Future Information Technology, 52056 Aachen, Germany Peter-Gr¨ unberg Institut and Institute for Advanced Simulation, Forschungszentrum J¨ ulich, 52425 J¨ ulich, Germany nanomat/QMAT/CESAM and Department of Physics, Universit´e de Lie` ge, 4000 Li`ege, Belgium Received 10 March 2020 / Received in final form 17 April 2020 Published online 3 June 2020 c EDP Sciences / Societ`
a Italiana di Fisica / Springer-Verlag GmbH Germany, part of Springer Nature, 2020 Abstract. We study the asymptotic decay of the Friedel density oscillations induced by an open boundary in a one-dimensional chain of lattice fermions with a short-range two-particle interaction. From TomonagaLuttinger liquid theory it is known that the decay follows a power law, with an interaction dependent exponent, which, for repulsive interactions, is larger than the noninteracting value −1. We first investigate if this behavior can be captured by many-body perturbation theory for either the Green function or the selfenergy in lowest order in the two-particle interaction. The analytic results of the former show a logarithmic divergence indicative of the power law. One might hope that the resummation of higher order terms inherent to the Dyson equation then leads to a power law in the perturbation theory for the self-energy. However, the numerical results do not support this. Next we use density functional theory within the local-density approximation and an exchange-correlation functional derived from the exact Bethe ansatz solution of the translational invariant model. While the numerical results are consistent with power-law scaling if systems of 104 or more lattice sites are considered, the extracted exponent is very close to the noninteracting value even for sizeable interactions.
1 Introduction The elementary excitations of one-dimensional (1d), metallic Fermi systems with a two-particle interaction are not given by fermionic quasi-particles, but are instead of collective, bosonic nature [1,2]. Such quantum many-body systems can thus not be described by Fermi liquid theory. For short-ranged, i.e. screened, two-particle interactions, on which we focus here, Tomonaga-Luttinger liquid theory is applicable instead [3]. One of the characteristics of Tomonaga-Luttinger liquids is the power-law decay of correlation functions at large times or spatial distances with exponents which, in spinless models, can be expressed in terms of a single parameter K. This Tomonaga-Luttinger liquid parameter depends on the band structure and filling as well as on the amplitude and range of the two-particle interaction of the model Hamiltonian. For repulsive interactions 0 < K < 1 while K > 1 for attractive ones; K = 1 corresponds to noninteracting f
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